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By Janet Hook, Wall Street Journal–

Donald Trump trounced his rivals up and down the Eastern Seaboard on Tuesday, winning five Republican presidential primaries that provide a shot of adrenaline for the home stretch of a campaign he has dominated from the start.

Mr. Trump was declared the winner in the Connecticut, Delaware, Maryland, Pennsylvania and Rhode Island primaries, a rout that leaves him close to having an insurmountable lead in his quest for the 1,237 delegates needed to nail down the GOP nomination.

“I consider myself the presumptive nominee, absolutely,” Mr. Trump said at a victory rally in New York. “We should heal the Republican Party.”

His five-state win was expected, but early returns suggested he would rack up some of his biggest winning margins of the year, even bigger in some states than the 60.4% victory he scored last week in New York.

Heading into Tuesday, Mr. Trump had 845 bound delegates, Texas Sen. Ted Cruz had 559 and Ohio Gov. John Kasich held 148. That meant Mr. Trump had to win about 65% of remaining bound delegates to reach the delegate target.

The exact number of delegates allocated to each candidate after Tuesday’s contests will depend in part on vote breakdowns in individual congressional districts. But Mr. Trump was in position to pick up at least 110 of the 118 bound delegates up for grabs in the five states.

Still, Mr. Trump in the coming weeks will have to fight through a series of potentially less-friendly states holding contests in May—Indiana, Nebraska, West Virginia, Oregon and Washington—and can’t reach that delegate threshold before the last day of primary voting June 7, when the five states voting include California.

His nearest rival, Mr. Cruz, was on track to place third in some if not most of the five states voting Tuesday. Even before the polls closed, Mr. Cruz was looking forward to Indiana and beyond. “Tonight this campaign moves back to more favorable terrain,” he said at a rally in Knightstown, Ind.

Mr. Kasich’s top strategist, John Weaver, also cautioned against a premature coronation. “Tonight proves one thing: this race will come down to June 7,” he said in a memo. “The truth is Donald Trump has no margin for error.”

Messrs. Cruz and Kasich still have a chance to collect enough delegates to block Mr. Trump from getting the 1,237 needed for a first-ballot victory at the Cleveland convention in July. But each week presents another potential Waterloo for the stop-Trump forces, who can’t afford another loss.

Our Principles PAC, an anti-Trump group, said Tuesday’s results were expected and their efforts would continue. “With Kasich and Cruz showing signs of cooperation and more challenging terrain ahead for Trump, this race is going all the way to Cleveland,” a group memo said.

Mr. Trump and his supporters have argued there would be a huge political backlash if he is the leading delegate- and vote-getter and is denied the nomination. Public opinion seems to be moving in his direction, as polls show increasing acceptance among Republicans for his candidacy.

“As far as I’m concerned, it’s over,” said Mr. Trump, who said he wasn’t planning to change his persona, as some of this campaign aides have suggested he would. He said he hadn’t yet begun considering who would be his running mate, but that he would set up a committee to begin the process “in the not too distant future.”

In the latest Wall Street Journal/NBC News poll, the share of GOP primary voters who say they would be satisfied with Mr. Trump as the nominee jumped to 63% in April, up 10 points in a month. And the share who could see themselves supporting him rose 5 points, to about equal with Messrs. Cruz and Kasich.

A weekly tracking poll by NBC News/Survey Monkey reported Tuesday that support for Mr. Trump among Republicans and GOP-leaning voters hit 50% for the first time since the survey started in late December. That finding and his commanding victories in the Tuesday primaries undercut the claim of his rivals that Mr. Trump has a ceiling of support that will keep him from garnering a majority in the party.

“I know people view him as crass, but I’m tired of the typical political nonsense,” said Dave Rothfield, 57, a retired radiologist and Trump supporter in Gaithersburg, Md. “Everybody portrays people who support him as uneducated slobs who don’t know what they’re talking about. But a lot of them are like me, people who follow this stuff closely and are tired of what they’re seeing.”

Mr. Trump may be benefiting from voter backlash against the Cruz and Kasich campaigns for saying this week they would coordinate campaign strategy to improve their chances of stopping the New York businessman’s momentum.

“I just think they are backstabbers,” said Phyllis Mooney, 75, a retiree in Danbury, Conn., who voted for Mr. Trump. “They are just doing what they have to do, regardless of what they say or whatever, to get Trump out.”

In Pennsylvania, Mr. Trump will pick up only 17 of the state’s 71 delegates. The other 54 will go to the convention not bound to any candidate, under the state’s decades-old system that leaves delegates free to vote as they choose.

The arrangement was confusing for voters because the state’s ballots listed delegate candidates with no indication if they supported a particular candidate.

The Trump and Cruz campaigns put out lists of Pennsylvania delegates it believed would be their supporters. One of those on the Trump list, Tom Uram, 56, said he was “all in” for Mr. Trump. The senior financial services representative for MetLife Inc. said he has been a delegate or an alternate delegate at all the Republican National Conventions since 1980.

He said he would cast a ballot for Mr. Trump “until the bitter end” if there were a contested convention. However, he predicted Mr. Trump would win enough delegates in the remaining primaries to secure the nomination.

“I think he’s going to hit the magic number on the first ballot, and I want to be there to cast the vote,” Mr. Uram said.

Mr. Kasich’s record in neighboring Ohio helped him pick up some votes in Pennsylvania.

“I look at Ohio, and John Kasich came in and really turned that state around,” said Ann Dugan, 61, former assistant dean of the University of Pittsburgh’s Joseph M. Katz Graduate School of Business. “We need to move away from beauty contests and hyperbole. I think we need a leader that understands the process and the art of the possible.”

At stake in Connecticut are 28 delegates, with 13 awarded to Mr. Trump as the statewide winner Three delegates also go to the winner of each of the state’s five congressional districts.

Delaware awards all 16 of its delegates to the candidate who wins the most statewide votes. The small mid-Atlantic state rarely draws much attention in presidential campaigns, but Mr. Trump held a large rally on Friday to energize his supporters.

Maryland awards 38 delegates, with 14 going to Mr. Trump for his statewide victory and three to the winner of each of its eight congressional districts. Mr. Kasich invested time there trying to connect with the state’s more centrist GOP voters.

Rhode Island allocates its 19 delegates proportionately, with 6 based on returns in the state’s two congressional districts and 13 based on statewide results. To qualify for any delegates, a candidate must win at least 10% of the vote.

—Reid J. Epstein, Patrick O’Connor, Kris Maher and Joseph De Avila
contributed to this article.